Generally we use definitions and notation as in [D]. In particular, (W,S) is a finitely generated Coxeter system, C is a building with Weyl group W , |C| is the Davis realization of C. We will, however, confuse the Coxeter group and its abstract Coxeter complex, denoting both by W ; in particular, |W | denotes the Davis complex. The W -valued distance in C will be denoted δC , while δ will be t...

Coxeter groups were introduced by Jacques Tits in the 1960s as a natural generalization of the groups generated by reflections which act geometrically (which means properly discontinuously cocompactly by isometries) on spheres and euclidean spaces. And ever since their introduction their basic structure has been reasonably well understood [BB05, Bou02, Dav08]. More precisely, every Coxeter grou...

A new recursive procedure of the calculation of partition numbers function W (s, d m) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W (s, d m) and their periods τ (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group S m are presented. A least common multiple lcm(m) of the series of the natural num...

Let Σ be the Davis complex for a Coxeter system (W, S). The automorphism group G of Σ is naturally a locally compact group, and a simple combinatorial condition due to Haglund– Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice Γ, and an infinite family of uniform lattices with covolume...

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph An−1 on n− 1 nodes. Here we describe an algebra depending on an arbitrary graph M , called the Brauer algebra of type M , and study its structure in...

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural “Coxeter element”. The folding of these graphs and groups is also discussed, using the theory of C-algebras.

We study the category O for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma’s result about homomorphisms between Verma modules.

In this paper, we show that the boundary ∂Σ(W, S) of a rightangled Coxeter system (W, S) is minimal if and only if W S̃ is irreducible, where W S̃ is the minimum parabolic subgroup of finite index in W . We also provide several applications and remarks.

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x)= k, for α∈Φ and k=0, 1, . . . ,m, is denoted here by AΦ . The characteristic polynomial of AΦ is related in the paper to that of the Coxeter arrangement AΦ (corresponding to m=0), and the number of regions into which the fun...

In this paper I describe a method – based on the projective interpretation of the hyperbolic geometry – that determines the data and the density of the optimal ball and horoball packings of each well-known Coxeter tiling (Coxeter honeycomb) in the hyperbolic space H.